Given a Banach algebra ℱ of complex-valued functions and a closed, linear (possibly unbounded) densely defined operator A, on a Banach space, with an ℱ functional calculus we present two ways of extending this functional calculus to a much larger class of functions with little or no growth conditions. We apply this to spectral operators of scalar type, generators of bounded strongly continuous groups and operators whose resolvent set contains a half-line. For f in this larger class, one construction measures how far f(A) is from generating a strongly continuous semigroup, while the other construction measures how far f(A) is from being bounded. We apply our constructions to evolution equations.
@article{bwmeta1.element.bwnjournal-article-smv114i3p237bwm,
author = {Ralph deLaubenfels},
title = {Automatic extensions of functional calculi},
journal = {Studia Mathematica},
volume = {113},
year = {1995},
pages = {237-259},
zbl = {0834.47012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv114i3p237bwm}
}
deLaubenfels, Ralph. Automatic extensions of functional calculi. Studia Mathematica, Tome 113 (1995) pp. 237-259. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv114i3p237bwm/
[00000] [1] M. Balabane, H. Emamirad and M. Jazar, Spectral distributions and generalization of Stone's theorem to the Banach space, Acta Appl. Math. 31 (1993), 275-295. | Zbl 0802.47013
[00001] [2] G. Da Prato, Semigruppi regolarizzabili, Ricerche Mat. 15 (1966), 223-248.
[00002] [3] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. | Zbl 0457.47030
[00003] [4] R. deLaubenfels, Unbounded holomorphic functional calculus for operators with polynomially bounded resolvents, J. Funct. Anal. 114 (1993), 348-394. | Zbl 0785.47018
[00004] [5] R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Math. 1570, Springer, 1994. | Zbl 0811.47034
[00005] [6] H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, 1978. | Zbl 0384.47001
[00006] [7] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Interscience, New York, 1971.
[00007] [8] E. Marschall, Functional calculi for closed linear operators in Banach spaces, Manuscripta Math. 35 (1981), 277-310. | Zbl 0496.47033
[00008] [9] L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, Pa., 1975. | Zbl 0302.35003
[00009] [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. | Zbl 0207.13501